In data assimilation problems, various types of data are naturally linked to different spatial resolutions (e.g., seismic and electromagnetic data), and these scales are usually not coincident to the subsurface simulation model scale. Alternatives like upscaling/downscaling of the data and/or the simulation model can be used, but with potential loss of important information. Such alternatives introduce additional uncertainties which are not in the nature of the problem description, but the result of the post processing of the data or the geo-model. To address this issue, a novel multiscale (MS) data assimilation method is introduced. The overall idea of the method is to keep uncertain parameters and observed data at their original representation scale, avoiding upscaling/downscaling of any quantity. The method relies on a recently developed mathematical framework to compute adjoint gradients via a MS strategy in an algebraic framework. The fine-scale uncertain parameters are directly updated and the MS grid is constructed in a resolution that meets the observed data resolution. This formulation therefore enables a consistent assimilation of data represented at a coarser scale than the simulation model. The misfit objective function is constructed to keep the MS nature of the problem. The regularization term is represented at the simulation model (fine) scale, whereas the data misfit term is represented at the observed data (coarse) scale. The computational aspects of the method are investigated in a simple synthetic model, including an elaborate uncertainty quantification step, and compared to upscaling/downscaling strategies. The experiment shows that the MS strategy provides several potential advantages compared to more traditional scale conciliation strategies: (1) expensive operations are only performed at the coarse scale; (2) the matched uncertain parameter distribution is closer to the “truth”; (3) faster convergence behavior occurs due to faster gradient computation; and (4) better uncertainty quantification results are obtained. The proof-of-concept example considered in this paper sheds new lights on how one can reduce uncertainty within fine-scale geo-model parameters with coarse-scale data, without the necessity of upscaling/downscaling the data nor the geo-model. The developments demonstrate how to consistently formulate such a gradient-based MS data assimilation strategy in an algebraic framework which allows for implementation in available computational platforms.

We introduce a semi-analytical iterative multiscale derivative computation methodology that allows for error control and reduction to any desired accuracy, up to fine-scale precision. The model responses are computed by the multiscale forward simulation of flow in heterogeneous porous media. The derivative computation method is based on the augmentation of the model equation and state vectors with the smoothing stage defined by the iterative multiscale method. In the formulation, we avoid additional complexity involved in computing partial derivatives associated to the smoothing step. We account for it as an approximate derivative computation stage. The numerical experiments illustrate how the newly introduced derivative method computes misfit objective function gradients that converge to fine-scale one as the iterative multiscale residual converges. The robustness of the methodology is investigated for test cases with high contrast permeability fields. The iterative multiscale gradient method casts a promising approach, with minimal accuracy-efficiency tradeoff, for large-scale heterogeneous porous media optimization problems.

A generic framework for the computation of derivative information required for gradient-based optimization using sequentially coupled subsurface simulation models is presented. The proposed approach allows for the computation of any derivative information with no modification of the mathematical framework. It only requires the forward model Jacobians and the objective function to be appropriately defined. The flexibility of the framework is demonstrated by its application in different reservoir management studies. The performance of the gradient computation strategy is demonstrated in a synthetic water-flooding model, where the forward model is constructed based on a sequentially coupled flow-transport system. The methodology is illustrated for a synthetic model, with different types of applications of data assimilation and life-cycle optimization. Results are compared with the classical fully coupled (FIM) forward simulation. Based on the presented numerical examples, it is demonstrated how, without any modifications of the basic framework, the solution of gradient-based optimization models can be obtained for any given set of coupled equations. The sequential derivative computation methods deliver similar results compared to FIM methods, while being computationally more efficient.

An efficient multiscale (MS) gradient computation method for subsurface flow management and optimization is introduced. The general, algebraic framework allows for the calculation of gradients using both the Direct and Adjoint derivative methods. The framework also allows for the utilization of any MS formulation that can be algebraically expressed in terms of a restriction and a prolongation operator. This is achieved via an implicit differentiation formulation. The approach favors algorithms for multiplying the sensitivity matrix and its transpose with arbitrary vectors. This provides a flexible way of computing gradients in a form suitable for any given gradient-based optimization algorithm. No assumption w.r.t. the nature of the problem or specific optimization parameters is made. Therefore, the framework can be applied to any gradient-based study. In the implementation, extra partial derivative information required by the gradient computation is computed via automatic differentiation. A detailed utilization of the framework using the MS Finite Volume (MSFV) simulation technique is presented. Numerical experiments are performed to demonstrate the accuracy of the method compared to a fine-scale simulator. In addition, an asymptotic analysis is presented to provide an estimate of its computational complexity. The investigations show that the presented method casts an accurate and efficient MS gradient computation strategy that can be successfully utilized in next-generation reservoir management studies.

A multiscale gradient computation method for multiphase flow in heterogeneous porous media is developed. The method constructs multiscale primal and dual coarse grids, imposed on the given fine-scale computational grid. Local multiscale basis functions are computed on (dual-) coarse blocks, constructing an accurate map (prolongation operator) between coarse- and fine-scale systems. While the expensive operations involved in computing the gradients are performed at the coarse scale, sensitivities with respect to uncertain parameters (e.g., grid block permeabilities) are expressed in the fine scale via the partial derivatives of the prolongation operator. Hence, the method allows for updating of the geological model, rather than the dynamic model only, avoiding upscaling and the inevitable loss of information. The formulation and implementation are based on automatic differentiation (AD), allowing for convenient extensions to complex physics. An IMPES coupling strategy for flow and transport is followed, in the forward simulation. The flow equation is computed using a multiscale finite volume (MSFV) formulation and the transport equation is computed at the fine scale, after reconstruction of mass conservative velocity field. To assess the performance of the method, a synthetic multiphase flow test case is considered. The multiscale gradients are compared against those obtained from a fine-scale reference strategy. Apart from its computational efficiency, the benefits of the method include flexibility to accommodate variables expressed at different scales, specially in multiscale data assimilation and reservoir management studies.